We normally count in bases, usually base ten. When you run out of the ten counting symbols or digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, you can carry on using them by doubling them up: 10, 11, 12, and when you run out of those then trebling up: 100, 101 . . . for as long as you live if you feel like it. You can count in quicker jumps too: from 7 to 13 by a jump known as adding the number 6, from 13 to 42 by adding 29. This jump is a bit more complicated because you have to carry a number, 1, at the same time.
But what if we had an infinite base? That means using a new symbol every time, no recycling old ones. So a very big quantity that takes maybe hundreds of digits to count up to in base ten would only need one in base infinity, say a smiley face. And all our jumps could be made without having to carry any number. So you can get from smiley face to say lightning bolt just by adding lollipop. Of course we might need a special publicly accessible book or database to keep all these digits in so everybody knows what you're talking about, but that wouldn't have to be actually infinite in size - just say several trillion in the early stages and continually growing as required. (So it's a bit like the Aristotelian potential infinity mentioned in the article) Every time somebody counted beyond the existing latest symbol they'd be legally or at least morally obliged to add the new symbol thus created to the database. Same as a copy of every printed book has to go to the Library of Congress.
What about the symbol for infinity then? Can it go in The Book? Yes, maybe but this time every new symbol would have to be entered before rather than after it. It could be a second order symbol for any as yet unadded counting symbol, rather like those three dots I used at the beginning . . . Or maybe not a book, but a book end.
We normally count in bases, usually base ten. When you run out of the ten counting symbols or digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, you can carry on using them by doubling them up: 10, 11, 12, and when you run out of those then trebling up: 100, 101 . . . for as long as you live if you feel like it. You can count in quicker jumps too: from 7 to 13 by a jump known as adding the number 6, from 13 to 42 by adding 29. This jump is a bit more complicated because you have to carry a number, 1, at the same time.
But what if we had an infinite base? That means using a new symbol every time, no recycling old ones. So a very big quantity that takes maybe hundreds of digits to count up to in base ten would only need one in base infinity, say a smiley face. And all our jumps could be made without having to carry any number. So you can get from smiley face to say lightning bolt just by adding lollipop. Of course we might need a special publicly accessible book or database to keep all these digits in so everybody knows what you're talking about, but that wouldn't have to be actually infinite in size - just say several trillion in the early stages and continually growing as required. (So it's a bit like the Aristotelian potential infinity mentioned in the article) Every time somebody counted beyond the existing latest symbol they'd be legally or at least morally obliged to add the new symbol thus created to the database. Same as a copy of every printed book has to go to the Library of Congress.
What about the symbol for infinity then? Can it go in The Book? Yes, maybe but this time every new symbol would have to be entered before rather than after it. It could be a second order symbol for any as yet unadded counting symbol, rather like those three dots I used at the beginning . . . Or maybe not a book, but a book end.